Not in the notes yet: Differentiation: rules (14.1-14.4), Tangents, normals and stationary points (14.5, 14.6, 14.9), Rates of change, small increments, practical max/min (14.7, 14.8), Kinematics (14.14, 14.15)

Formula bank

Every formula and rule in this chapter. All Additional Mathematics formulas on one page →

Integration: powers, 1/x and (ax + b)ⁿ

PowersLearn it\(\displaystyle\int x^{n}\,dx=\frac{x^{n+1}}{n+1}+c\)add 1 to the power, divide by the NEW power. Any n except −1.Use it when every power of x, roots, 1/x²…

A constantLearn it\(\displaystyle\int k\,dx=kx+c\)Use it when a number on its own.

One over xLearn it\(\displaystyle\int \frac1x\,dx=\ln x+c\)the n = −1 case.Use it when \(\tfrac1x\), \(\tfrac5x\).

Bracket to a powerLearn it\(\displaystyle\int (ax+b)^{n}\,dx=\frac{(ax+b)^{n+1}}{a(n+1)}+c\)raise the power, divide by the new power AND by a.Use it when \((3x-1)^5\), \(\sqrt{4x+1}\).

One over a bracketLearn it\(\displaystyle\int \frac{1}{ax+b}\,dx=\frac1a\ln(ax+b)+c\)Use it when \(\tfrac{4}{2x+5}\).

Integrating exponentials and trig

ExponentialLearn it\(\displaystyle\int e^{ax+b}\,dx=\frac1a e^{ax+b}+c\)e stays e; divide by a.Use it when \(e^{2x}\), \(e^{1-2x}\).

CosineLearn it\(\displaystyle\int\cos(ax+b)\,dx=\frac1a\sin(ax+b)+c\)Use it when cos.

SineLearn it\(\displaystyle\int\sin(ax+b)\,dx=-\frac1a\cos(ax+b)+c\)note the minus.Use it when sin.

Sec squaredLearn it\(\displaystyle\int\sec^{2}(ax+b)\,dx=\frac1a\tan(ax+b)+c\)Use it when \(\sec^{2}\).

IdentitiesLearn it\(\tan^{2}x=\sec^{2}x-1\); \(\sin x\cos x=\tfrac12\sin2x\); \((\sin x+\cos x)^{2}=1+\sin2x\)Use it when \(\tan^{2}\), products of sin and cos.

Definite integrals and unknown limits

Definite integralLearn it\(\displaystyle\int_a^b f(x)\,dx=\big[F(x)\big]_a^b=F(b)-F(a)\)F = any integral of f; no + c (it cancels).Use it when limits on the integral sign.

Swap the limitsLearn it\(\displaystyle\int_b^a f\,dx=-\int_a^b f\,dx\)Use it when limits the 'wrong' way round.

Log lawsLearn it\(\ln p+\ln q=\ln pq\); \(\ln p-\ln q=\ln\frac pq\); \(k\ln p=\ln p^{k}\)Use it when 'as a single logarithm', 'in the form ln a'.

Exact valuesLearn it\(\ln1=0\), \(e^{0}=1\), \(e^{\ln k}=k\), \(\sin\tfrac{\pi}{6}=\tfrac12\), \(\tan\tfrac{\pi}{4}=1\)Use it when Paper 1 (no calculator).

Differentiate, then integrate ('hence')

The key factLearn itIf \(\dfrac{d}{dx}F(x)=g(x)\) then \(\displaystyle\int g(x)\,dx=F(x)+c\)Use it when 'hence', 'use your answer to part (a)'.

A multipleLearn it\(\displaystyle\int k\,g(x)\,dx=k\,F(x)+c\)Use it when the integrand is a number × the derivative.

Extra termsLearn itIf \(\frac{d}{dx}F=g+h\) and you can integrate h: \(\displaystyle\int g\,dx=F-\int h\,dx\)Use it when the derivative has an extra easy term.

Product ruleLearn it\(\dfrac{d}{dx}(uv)=u\dfrac{dv}{dx}+v\dfrac{du}{dx}\)you will usually need it in part (a).Use it when x ln x, x e^(kx), x sin kx.

Finding a curve from its gradient

From the gradientLearn it\(y=\displaystyle\int\frac{dy}{dx}\,dx\), then use the point to find cUse it when 'find the equation of the curve'.

From the second derivativeLearn it\(\frac{dy}{dx}=\int\frac{d^{2}y}{dx^{2}}dx\) (+ c₁), then \(y=\int\frac{dy}{dx}dx\) (+ c₂)the gradient fact fixes c₁, the point fixes c₂.Use it when \(\frac{d^{2}y}{dx^{2}}\) or f″(x) given.

Normal to tangentLearn it\(m_{\text{tangent}}=-\dfrac{1}{m_{\text{normal}}}\)Use it when 'gradient of the normal is …'.

Stationary pointLearn it\(\frac{dy}{dx}=0\)use the GIVEN derivative, then the curve for y.Use it when 'find the stationary point'.

Areas under curves

Area above the axisLearn it\(\text{Area}=\displaystyle\int_a^b y\,dx\)curve above the x-axis between a and b.Use it when 'area under the curve'.

Area below the axisLearn it\(\text{Area}=\left|\displaystyle\int_a^b y\,dx\right|\)the integral comes out negative; the area is its size.Use it when region under the x-axis.

Crossing the axisLearn itsplit at each root; add the SIZES of the piecesUse it when region on both sides of the axis.

Sum of two areasLearn it\(\displaystyle\int_a^c f\,dx+\int_c^b g\,dx\)different curves bound different parts.Use it when the boundary changes at a point.

Areas between lines and curves

Top − bottomLearn it\(\text{Area}=\displaystyle\int_a^b(y_{\text{top}}-y_{\text{bottom}})\,dx\)a and b = where the two graphs meet (or given lines).Use it when region between two graphs.

TriangleLearn it\(\tfrac12\times\text{base}\times\text{height}\)Use it when a straight edge down to the x-axis.

TrapeziumLearn it\(\tfrac12(a+b)h\)the parallel sides are the two vertical heights.Use it when a line above the x-axis between two x values.

Tangent / normalLearn it\(y-y_1=m(x-x_1)\); normal gradient \(=-\tfrac1m\)Use it when region bounded by a tangent or normal.

Summary notes

The key points of every note, for a last read before a test.